Monogenic trinomials of the form $x^4+ax^3+d$ and their Galois groups

Abstract

Let $f(x)=x^4+ax^3+d\in {\mathbb Z}[x]$, where $ad\ne 0$. Let $C_n$ denote the cyclic group of order $n$, $D_4$ the dihedral group of order 8, and $A_4$ the alternating group of order 12. Assuming that $f(x)$ is monogenic, we give necessary and sufficient conditions involving only $a$ and $d$ to determine the Galois group $G$ of $f(x)$ over ${\mathbb Q}$. In particular, we show that $G=D_4$ if and only if $(a,d)=(\pm 2,2)$, and that $G\not \in \{C_4,C_2\times C_2\}$. Furthermore, we prove that $f(x)$ is monogenic with $G=A_4$ if and only if $a=4k$ and $d=27k^4+1$, where $k\ne 0$ is an integer such that $27k^4+1$ is squarefree. This article extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups.